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Chapter 2
Mathematics of
Cryptography
Part I: Modular Arithmetic, Congruence,
and Matrices
2.1
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Objectives
 To review integer arithmetic, concentrating on
divisibility and finding the greatest common
divisor using the Euclidean algorithm
 To understand how the extended Euclidean
algorithm can be used to solve linear
Diophantine equations, to solve linear
congruent equations, and to find the
multiplicative inverses
2.2
Objectives
 To emphasize the importance of modular
arithmetic and the modulo operator, because
they are extensively used in cryptography
 To emphasize and review matrices and
operations on residue matrices that are
extensively used in cryptography
 To solve a set of congruent equations using
residue matrices
2.3
2.1
INTEGER ARITHMETIC
In integer arithmetic,
we use a set and a few operations.
You are familiar with this set and
the corresponding operations,
but they are reviewed here to create
a background for modular arithmetic.
2.4
2.1.1 Set of Integers
The set of integers, denoted by Z, contains all
integral numbers (with no fraction) from negative
infinity to positive infinity (Figure 2.1).
Figure 2.1 The set of integers
2.5
2.1.2 Binary Operations
In cryptography, we are interested
in three binary operations applied
to the set of integers.
A binary operation takes two inputs and
creates one output.
Figure 2.2 Three binary operations for the set of integers
2.6
2.1.2 Binary Operations
The following shows the results of
the three binary operations on two integers.
Because each input can be
either positive or negative,
we can have four cases for each operation.
Example 2.1
2.7
2.1.3 Integer Division
In integer arithmetic, if we divide a by n,
we can get q and r .
The relationship between these four integers
can be shown as
a=q×n+r
2.8
2.1.3 Integer Division
Example 2.2
Assume that a = 255 and n = 11.
We can find q = 23 and r = 2 using the division algorithm.
Figure 2.3 Example 2.2, finding the quotient and the remainder
2.9
2.1.3 Integer Division
Figure 2.4 Division algorithm for integers
2.10
2.1.3 Integer Division
Example 2.3
When we use a computer or a calculator,
r and q are negative when a is negative.
How can we apply the restriction
that r needs to be positive?
The solution is simple, we decrement the value of q by 1
and we add the value of n to r to make it positive.
2.11
2.1.3 Integer Division
Figure 2.5 Graph of division algorithm
2.12
2.1.4 Divisibility
If a is not zero and we let r = 0 in the division relation,
we get
a=q×n
If the remainder is zero,
If the remainder is not zero,
2.13
2.1.4 Divisibility
Example 2.4
a. The integer 4 divides the integer 32
because 32 = 8 × 4.
We show this as
b. The number 8 does not divide the number 42
because 42 = 5 × 8 + 2.
There is a remainder, the number 2, in the equation.
We show this as
2.14
2.1.4 Divisibility
Properties
Property 1: if a|1, then a = ±1.
Property 2: if a|b and b|a, then a = ±b.
Property 3: if a|b and b|c, then a|c.
Property 4: if a|b and a|c, then
a|(m × b + n × c),
where m and n are arbitrary integers.
2.15
2.1.4 Divisibility
Example 2.5
2.16
2.1.4 Divisibility
Example 2.6
2.17
2.1.4 Divisibility
Note
Fact 1: The integer 1 has only one divisor, itself.
Fact 2: Any positive integer has
at least two divisors, 1 and itself
(but it can have more).
2.18
2.1.4 Divisibility
Figure 2.6 Common divisors of two integers
2.19
2.1.4 Divisibility
Greatest Common Divisor
The greatest common divisor of
two positive integers
is the largest integer that can divide both integers.
Euclidean Algorithm
Fact 1: gcd (a, 0) = a
Fact 2: gcd (a, b) = gcd (b, r), where r is
the remainder of dividing a by b
2.20
2.1.4 Divisibility
Figure 2.7 Euclidean Algorithm
When gcd (a, b) = 1,
we say that a and b are relatively prime.
2.21
2.1.4 Divisibility
Example 2.7
Find the greatest common divisor of 2740 and 1760.
Solution
We have gcd (2740, 1760) = 20.
2.22
2.1.4 Divisibility
Example 2.8
Find the greatest common divisor of 25 and 60.
Solution
We have gcd (25, 65) = 5.
2.23
2.1.4 Divisibility
Extended Euclidean Algorithm
Given two integers a and b,
we often need to find other two integers, s and t,
such that
The extended Euclidean algorithm can
calculate the gcd (a, b) and
at the same time calculate the value of s and t.
2.24
2.1.4 Divisibility
Figure 2.8.a Extended Euclidean algorithm, part a
where r = r1 - q×r2, s = s1 - q×s2, t = t1 - q×t2.
2.25
2.1.4 Divisibility
Figure 2.8.b Extended Euclidean algorithm, part b
2.26
2.1.4 Divisibility
Example 2.9
Given a = 161 and b = 28,
find gcd (a, b) and the values of s and t.
Solution
We get gcd (161, 28) = 7, s = −1 and t = 6.
where r = r1 - q×r2, s = s1 - q×s2, t = t1 - q×t2.
2.27
2.1.4 Divisibility
Example 2.10
Given a = 17 and b = 0,
find gcd (a, b) and the values of s and t.
Solution
We get gcd (17, 0) = 17, s = 1, and t = 0.
2.28
2.1.4 Divisibility
Example 2.11
Given a = 0 and b = 45,
find gcd (a, b) and the values of s and t.
Solution
We get gcd (0, 45) = 45, s = 0, and t = 1.
2.29
2.1.4 Divisibility
Linear Diophantine Equation
A linear Diophantine equation of two variables is
ax + by = c.
This equation has either no solution
or an infinite number of solutions.
Let d = gcd(a, b).
If d | c, the equation has no solution.
If d|c, the equation has an infinite number of solutions.
2.30
2.1.4 Divisibility
Linear Diophantine Equation
( ax + by = c when d|c )
Particular solution:
1. Reduce the equation to a1x + b1y = c1
by dividing both sides by d.
2. Solve s and t in the relation a1s + b1t = 1
using the extended Euclidean algorithm.
3. The particular solution can be found:
x0 = (c/d)s and
2.31
y0 = (c/d)t
2.1.4 Divisibility
General solutions:
x = x0 + k (b/d) and y = y0 − k(a/d)
where k is an integer
2.32
2.1.4 Divisibility
Example 2.12
Find the particular and general solutions to the equation
21x + 14y = 35.
Solution
2.33
2.1.4 Divisibility
Example 2.13
Imagine we want to cash a $100 check
and get some $20 and some $5 bills.
We have many choices, which we can find by solving
the corresponding Diophantine equation 20x + 5y = 100.
Since d = gcd (20, 5) = 5 and 5 | 100,
the equation has an infinite number of solutions,
but only a few of them are acceptable in this case.
The general solutions with x and y nonnegative are
(0, 20), (1, 16), (2, 12), (3, 8), (4, 4), (5, 0).
2.34
2.2 MODULAR ARITHMETIC
The division relationship (a = q × n + r) discussed
in the previous section
has two inputs (a and n) and two outputs (q and r).
In modular arithmetic, we are interested in
only one of the outputs, the remainder r.
2.35
2.2.1 Modulo Operator
The modulo operator is shown as mod.
The second input (n) is called the modulus.
The output r is called the residue.
Figure 2.9 Division algorithm and modulo operator
2.36
2.2.1 Modulo Operator
Example 2.14
Find the result of the following operations:
a. 27 mod 5
b. 36 mod 12
c. −18 mod 14
d. −7 mod 10
Solution
a. Dividing 27 by 5 results in r = 2
b. Dividing 36 by 12 results in r = 0.
c. Dividing −18 by 14 results in r = −4.
After adding the modulus to r = 10.
d. Dividing −7 by 10 results in r = −7.
After adding the modulus to −7, r = 3.
2.37
2.2.2 Set of Residues
The modulo operation creates a set,
which in modular arithmetic is referred to as
the set of least residues modulo n, or Zn.
Figure 2.10 Some Zn sets
2.38
2.2.3 Congruence
To show that two integers are congruent,
we use the congruence operator ( ≡ ).
For example, we write:
2.39
2.2.3 Congruence
Figure 2.11 Concept of congruence
2.40
2.2.3 Congruence
Residue Classes
A residue class [a] or [a]n is
the set of integers congruent modulo n.
2.41
2.2.3 Congruence
Figure 2.12 Comparison of Z and Zn using graphs
2.42
2.2.3 Congruence
Example 2.15
We use modular arithmetic in our daily life;
for example, we use a clock to measure time.
Our clock system uses modulo 12 arithmetic.
However, instead of a 0 we use the number 12.
2.43
2.2.4 Operation in Zn
The three binary operations that we discussed for the set Z
can also be defined for the set Zn.
The result may need to be mapped to Zn
using the mod operator.
Figure 2.13 Binary operations in Zn
2.44
2.2.4 Operation in Zn
Example 2.16
Perform the following operations (the inputs come from Zn):
a. Add 7 to 14 in Z15.
b. Subtract 11 from 7 in Z13.
c. Multiply 11 by 7 in Z20.
Solution
2.45
2.2.4 Operation in Zn
Example 2.17
Perform the following operations
(the inputs come from either Z or Zn):
a. Add 17 to 27 in Z14.
b. Subtract 43 from 12 in Z13.
c. Multiply 123 by −10 in Z19.
Solution
2.46
2.2.4 Operation in Zn
Properties
2.47
2.2.4 Operation in Zn
Figure 2.14 Properties of mode operator
2.48
2.2.4 Operation in Zn
Example 2.18
The following shows the application
of the above properties:
1. (1,723,345 + 2,124,945) mod 11 = (8 + 9) mod 11 = 6
2. (1,723,345 − 2,124,945) mod 11 = (8 − 9) mod 11 = 10
3. (1,723,345 × 2,124,945) mod 11 = (8 × 9) mod 11 = 6
2.49
2.2.4 Operation in Zn
Example 2.19
In arithmetic,
we often need to find the remainder of powers of 10
when divided by an integer.
2.50
2.2.4 Operation in Zn
Example 2.20
We have been told in arithmetic
that the remainder of an integer divided by 3
is the same as the remainder of the sum of
its decimal digits.
We write an integer as the sum of its digits
multiplied by the powers of 10.
2.51
2.2.5 Inverses
When we are working in modular arithmetic,
we often need to find the inverse of a number
relative to an operation.
We are normally looking for
an additive inverse (relative to an addition operation)
or a multiplicative inverse
(relative to a multiplication operation).
2.52
2.2.5 Inverses
Additive Inverse
In Zn, two numbers a and b are
additive inverses of each other if
In modular arithmetic,
each integer has an additive inverse.
The sum of an integer and its additive inverse
is congruent to 0 modulo n.
2.53
2.2.5 Inverses
Example 2.21
Find all additive inverse pairs in Z10.
Solution
The six pairs of additive inverses are
(0, 0), (1, 9), (2, 8), (3, 7), (4, 6), and (5, 5).
2.54
2.2.5 Inverses
Multiplicative Inverse
In Zn, two numbers a and b are
the multiplicative inverse of each other if
In modular arithmetic,
an integer may or may not have a multiplicative inverse.
When it does,
the product of the integer and its multiplicative inverse
is congruent to 1 modulo n.
2.55
2.2.5 Inverses
Example 2.22
Find the multiplicative inverse of 8 in Z10.
Solution
There is no multiplicative inverse because gcd (10, 8) = 2 ≠ 1.
In other words, we cannot find any number between 0 and 9
such that when multiplied by 8, the result is congruent to 1.
Example 2.23
Find all multiplicative inverses in Z10.
Solution
There are only three pairs: (1, 1), (3, 7) and (9, 9).
The numbers 0, 2, 4, 5, 6, and 8
do not have a multiplicative inverse.
2.56
2.2.5 Inverses
Example 2.24
Find all multiplicative inverse pairs in Z11.
Solution
We have seven pairs:
(1, 1), (2, 6), (3, 4), (5, 9), (7, 8), (9, 9), and (10, 10).
2.57
2.2.5 Inverses
The extended Euclidean algorithm finds
the multiplicative inverses of b in Zn
when n and b are given and gcd (n, b) = 1.
The multiplicative inverse of b is the value of t
after being mapped to Zn.
2.58
2.2.5 Inverses
Figure 2.15 Using extended Euclidean algorithm
to find multiplicative inverse
2.59
2.2.5 Inverses
Example 2.25
Find the multiplicative inverse of 11 in Z26.
Solution
The gcd (26, 11) is 1; the inverse of 11 is -7 or 19.
2.60
2.2.5 Inverses
Example 2.26
Find the multiplicative inverse of 23 in Z100.
Solution
The gcd (100, 23) is 1; the inverse of 23 is -13 or 87.
2.61
2.2.5 Inverses
Example 2.27
Find the inverse of 12 in Z26.
Solution
The gcd (26, 12) is 2; the inverse does not exist.
2.62
2.2.6 Addition and
Multiplication Tables
Figure 2.16 Addition and multiplication table for Z10
2.63
2.2.7 Different Sets
We need to use Zn
when additive inverses are needed;
we need to use Zn*
when multiplicative inverses are needed.
Figure 2.17 Some Zn and Zn* sets
2.64
2.2.8 Two More Sets
Cryptography often uses two more sets:
Zp and Zp*.
The modulus p in these two sets is a prime number.
2.65
2.3 MATRICES
In cryptography we need to handle matrices.
Although this topic belongs to
a special branch of algebra called linear algebra,
the following brief review of matrices is
necessary preparation
for the study of cryptography.
2.66
2.3.1 Definition
Figure 2.18 A matrix of size l x m
2.67
2.3.1 Definition
Figure 2.19 Examples of matrices
2.68
2.3.2 Operations and Relations
Equality, Addition and Subtraction.
Example 2.28
Figure 2.20 shows an example of
addition and subtraction.
Figure 2.20 Addition and subtraction of matrices
2.69
2.3.2 Operations and Relations
Multiplication.
Example 2. 29
Figure 2.21 shows the product of
a row matrix (1 × 3) by a column matrix (3 × 1).
The result is a matrix of size 1 × 1.
Figure 2.21 Multiplication of a row matrix by a column matrix
2.70
2.3.2 Operations and Relations
Example 2. 30
Figure 2.22 shows the product of
a 2 × 3 matrix by a 3 × 4 matrix.
The result is a 2 × 4 matrix.
Figure 2.22 Multiplication of a 2 × 3 matrix by a 3 × 4 matrix
2.71
2.3.2 Operations and Relations
Scalar Multiplication.
Example 2. 31
Figure 2.23 shows an example of scalar multiplication.
Figure 2.23 Scalar multiplication
2.72
2.3.3 Determinant
The determinant of a square matrix A of size m × m
denoted as det (A)
is a scalar calculated recursively as shown below:
The determinant is defined only for a square matrix.
2.73
2.3.3 Determinant
Example 2. 32
Figure 2.24 shows how we can calculate
the determinant of a 2 × 2 matrix
based on the determinant of a 1 × 1 matrix.
Figure 2.24 Calculating the determinant of a 2 × 2 matrix
2.74
2.3.3 Determinant
Example 2. 33
Figure 2.25 shows the calculation of
the determinant of a 3 × 3 matrix.
Figure 2.25 Calculating the determinant of a 3 × 3 matrix
2.75
2.3.4 Inverses
Additive inverse : A + B = 0.
Multiplicative inverse : A × B = B × A = I.
Multiplicative inverses are only defined for square matrices.
Multiplicative inverse exists only if the det(A) has
a multiplicative inverse in the corresponding set.
2.76
2.3.5 Residue Matrices
Cryptography uses residue matrices:
matrices where all elements are in Zn.
A residue matrix has a multiplicative inverse
if gcd (det(A), n) = 1.
2.77
2.3.5 Residue Matrices
Example 2. 34
Figure 2.26 A residue matrix and its multiplicative inverse in Z26
Scalar Multiplication.
2.78
2.4 LINEAR CONGRUENCE
Cryptography often involves
solving an equation or
a set of equations of one or more variables
with coefficient in Zn.
This section shows how to solve equations
when the power of each variable is 1
(linear equation).
2.79
2.4.1 Single-Variable Linear Equations
Equations of the form ax ≡ b (mod n ) might have
no solution or a limited number of solutions.
If d|b, we use the following strategy to find solutions:
1. Reduce the equation by dividing both sides by d.
2. Multiply both sides of the reduced equation
by multiplicative inverse of a (including the modulus)
to find the particular solution x0.
3. The general solutions are x = x0 + k(n/d)
for k = 0, 1, 2, …, (d-1).
2.80
2.4.1 Single-Variable Linear Equations
Example 2.35
Solve the equation 10 x ≡ 2(mod 15).
Solution
First we find the gcd (10 and 15) = 5.
Since 5 does not divide 2, we have no solution.
Example 2.36
Solve the equation 14 x ≡ 12 (mod 18).
Solution
2.81
2.4.1 Single-Variable Linear Equations
Example 2.37
Solve the equation 3x + 4 ≡ 6 (mod 13).
Solution
First we change the equation to the form ax ≡ b (mod n).
We add −4 (the additive inverse of 4) to both sides,
which give 3x ≡ 2 (mod 13).
Because gcd (3, 13) = 1, the equation has only one solution,
which is x0 = (2 × 3−1) mod 13 = 18 mod 13 = 5.
We can see that the answer satisfies the original equation:
3 × 5 + 4 ≡ 6 (mod 13).
2.82
2.4.2 Single-Variable Linear Equations
We can also solve a set of linear equations with the
same modulus if the matrix formed from the
coefficients of the variables is invertible.
Figure 2.27 Set of linear equations
2.83
2.4.2 Single-Variable Linear Equations
Example 2.38
Solve the set of following three equations:
Solution
The result is
x ≡ 15 (mod 16), y ≡ 4 (mod 16), and z ≡ 14 (mod 16).
We can check the answer
by inserting these values into the equations.
2.84